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<h1 id="the-math-of-survival-analysis">The math of Survival Analysis</h1>
<p>Now that we <a href="intro.html">have introduced the main notions in Survival Analysis</a>, let's define the variables and functions that we will be using and give simple examples to provide additional insight:</p>
<hr />
<h2 id="t-survival-time">
<script type="math/tex">T</script>, Survival Time</h2>
<p>
<script type="math/tex">T</script> is a positive random variable representing the waiting time until an event occurs. Its probability density function (p.d.f.) is <script type="math/tex">f(t)</script> and cumulative distribution function (c.d.f.) is given by
<script type="math/tex; mode=display">\begin{equation*}
F(t) = \text{Pr} \left[ T < t \right] = \int_{-\infty}^{t} f(u) du
\end{equation*}</script>
</p>
<ul>
<li><strong>Example:</strong>
Let's take the example of <a href="tutorials/credit_risk.html">credit risk</a> and assume that the event of interest is <code>fully repaying a loan</code>. We can now analyze the cumulative distribution function of two distinct borrowers through time.
<center><table class="image">
<caption align="bottom">Figure 1 - Comparing cumulative distribution functions</caption>
<tr><td><center><img src="images/math_cdf.png" alt="PySurvival - Comparing cumulative distribution functions" title="PySurvival - Comparing cumulative distribution functions" width=100%, height=100%  /></center>
</td></tr>
</table>
</center>
Here, we can see that the probability that Borrower B has fully repaid his/her loan reaches 50% or 80% much faster than Borrower A's. This indicates that Borrower B is potentially less risky than Borrower A.</li>
</ul>
<hr />
<h2 id="st-x-survival-function">
<script type="math/tex">S(t, x)</script>, Survival function</h2>
<p>
<script type="math/tex">S(t)</script> is the probability that the event of interest has not occurred by some time <script type="math/tex">t</script>
<script type="math/tex; mode=display">\begin{equation*}
S(t) = 1 - F(t) = \text{Pr} \left[ T \geq t\right]
\end{equation*}</script>
</p>
<ul>
<li>
<p><strong>Example:</strong>
Here, we will consider the example of <a href="tutorials/churn.html">churn modeling</a>, assume that the event of interest is <code>stopping the SaaS subscription</code> and analyze the survival function of three distinct customers through time.
<center><table class="image">
<caption align="bottom">Figure 2 - Comparing survival functions</caption>
<tr><td><center><img src="images/math_surv.png" alt="PySurvival - Comparing survival functions" title="PySurvival - Comparing survival functions" width=100%, height=100%  /></center>
</td></tr>
</table>
</center>
Here, we can see that the probability of remaining a customer reaches 50% much faster for Client C than Client B. On the other hand, Client A's  probability doesn't even go below 60% from week 0 to week 15 of the analysis. In a nutshell, </p>
<ul>
<li>Client C is very likely to churn within the first 2 weeks</li>
<li>Client B is likely to churn within the next 15 weeks</li>
<li>Client A is very likely to remain a customer within the next 15 weeks.</li>
</ul>
</li>
</ul>
<hr />
<h2 id="ht-x-hazard-function-and-rx-risk-score">
<script type="math/tex">h(t, x)</script>, hazard function and <script type="math/tex">r(x)</script> risk score</h2>
<p>
<script type="math/tex">h(t)</script> expresses the conditional probability that the event will occur within <script type="math/tex">[t, t+dt)</script> , given that it has not occurred before.
<script type="math/tex; mode=display">\begin{equation*}
h(t) = \lim_{dt \to 0} \frac{\text{Pr} \left[  t \leq T < t + dt\ | T \geq t\right ] }{dt} = \frac{f(t)}{S(t)} = -\frac{d}{dt}\log  S(t)
\end{equation*}</script>
</p>
<p>Thus, the hazard and Survival functions are linked by the following formula:
<script type="math/tex; mode=display">\begin{equation*}
S(t) = \exp\left(- \int_{0}^{t} h(u) du \right)
\end{equation*}</script>
where <script type="math/tex">H(t) = \int_{0}^{t} h(u) du </script> is the cumulative hazard function</p>
<p>However, the hazard function is rarely used in its original form. Most of the time, we subdivide the time axis in <script type="math/tex">J</script> parts and calculate the risk score of a sample <script type="math/tex">x</script>, such that:
<script type="math/tex; mode=display">\begin{equation*}
r(x) = \sum_{j=1}^J H(t_j, x)
\end{equation*}</script>
</p>
<ul>
<li><strong>Example:</strong> Let's reuse our <strong>churn</strong> example. The previous conclusion can be translated into risk scores such that: <script type="math/tex">r(x_A) < r(x_B) < r(x_C)</script>. Indeed the faster you experience the event, the higher your risk score is.</li>
</ul>









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